Transcript Linear Equations in 3D Space

Experimental Mathematics
25 August 2011
Linear Transformations
and Ranks
Recall from Linear Algebra I:
Let V and W be vector spaces. A linear transformation
T :V W
is a function with the following properties:
(i ) T (u  v)  T (u )  T (v) for all u, v V
(ii )
T (cv)  cT (v)
for all v V , c  R
Basic example of a linear transformation
An m  n matrix A determines a linear tra nsformatio n
T : Rn  Rm
given by
T (x)  Ax
Visualizing Vectors
To visualize linear transformations, we need to be able
to visualize vectors first. Try the following in Sage:
v = vector([1,2])
w = vector([4,1])
plot(v) + plot(w) + plot(v - w)
The Sage function on the next slide plots the vectors in a
list L in different colors (it is not necessary to understand
the details of the function, we just use it as a tool).
Vector Plotting Function
def vector_plotter(vec_list, points=False):
l = float(len(vec_list))
pic = Graphics()
for i, v in enumerate(vec_list):
if points:
pic += point(v,hue=(i/l))
else:
pic += plot(v,hue=(i/l))
return pic
Example1
L=[vector([1,0]),vector([0,1]),vector([-1,0]),vector([0,-1])]
vector_plotter(L)
Example 2
L=[vector([1,0]),vector([0.75,0.25]),vector([0.5,0.5]),vector([0.25,0.75])]
vector_plotter(L,points=true)
Note: points=true has the effect that vectors are represented by points
instead of arrows.
Visualization of
Linear Transformations
We will visualize 2D linear transformations
by applying them to vectors in the plot on
the following slide.
Example 3
L = [vector([cos(i),sin(i)]) for i in srange(0,2*pi,2*pi/20,universe=RDF)]
vector_plotter(L)
Note: srange(a,b,c) returns the list of numbers in [a,b) starting with a and with
step size c. universe=RDF makes sure the number are in double precision
floating point format.
Problem 1
Modify Example 3 so that an ellipse with
axes of length 2 and 3 is shown.
The map Function
Python has a built in map function which can be used to apply a function f to
all entries of a list L. Syntax: map(f,L)
Examples:
L=range(5)
def f(x):
return x^2
map(f,L)
# output [0,1,4,9,16]
Multiply all vectors in a list with a matrix:
L=[vector([1,0]),vector([0,1])]
M=matrix([[3,5],[1,2]])
def f(v):
return M*v
map(f,L)
Problem 2
• Let L be the list of vectors from Example 3.
Use the map function to multiply all vectors
in L with the matrix [[1,0.3],[0.3,0.8]] and
plot the resulting list using the
vector_plotter function.
• Try this for other matrices. What are the
possible “shapes” that can arise?
Recall from Linear Algebra I:
Rank of a matrix A
 maximum number of linearly independen t rows of A
 maximum number of linearly independen t columns of A
 number of nonzero rows in a row - echelon form a A
kernel of A  set of all vectors x with Ax  0
Problem 3
• Try the matrix [[1,0.3],[2,0.6]] in Problem
2.
• Find the (unique) matrix for Problem 2 for
which the plot collapses to a single point.
• What is the connection between the rank of
the matrices and the “shape” of the plot?
Example 4:
Plot a Random Sample of Vectors (as Points)
L = [vector([random(),random()]) for i in range(20)]
vector_plotter(L, points=True)
Problem 4
• Let L be the list of vectors from Example 4.
Use the map function to multiply all vectors
in L with the matrix [[1,0.3],[2,0.6]] and
plot the resulting list using the
vector_plotter function (with points=true).
• Observe carefully exactly where each
original point lands. Is it random?
Example 5:
Plot Vectors on a Certain Line
L = [random()*vector([3,-10]) for i in range(100)]
vector_plotter(L, points=True)
Problem 5
a. Let L be the list of vectors from Example 4. Use the map
function to multiply all vectors in L with the matrix
[[1,0.3],[2,0.6]] and plot the resulting list using the
vector_plotter function (with points=true).
b. What happens? Note: Numbers of absolute value <1e-15
can be considered as 0 (occur due to round off errors)
c. Compute the kernel of the matrix [[1,0.3],[2,0.6]]. Note:
for a matrix A the sage-command A.right_kernel() returns
a representation of the kernel of A.
d. What is the connection between the kernel of
[[1,0.3],[2,0.6]] and the plot from part a?
Problem 6
The following generates random 4x5 matrices and
computes their rank and kernel:
m = random_matrix(QQ, 4, 5, algorithm='echelonizable', rank=randint(0,4),
upper_bound=60)
print m
print m.rank()
m.right_kernel()
Repeat this several times and especially take note of the
connection between the rank and the dimension of the
kernel. What is the mathematical theorem behind this?